Double-Sided Synchronizer

Mechanical double-sided synchronizer

Libraries:
Simscape / Driveline / Clutches

Description

The Double-Sided Synchronizer block represents a double-sided synchronizer that contains two back-to-back dog clutches, two back-to-back cone clutches, and one translational detent. In contrast to regular synchronizers, double-sided synchronizers synchronize and mesh gears on both the input and output sides of a transmission. Double-sided synchronizers can engage and disengage gears on two shafts simultaneously, which allows for more complex gear engagement patterns. Heavy-duty commercial vehicles, off-highway equipment, and some high-performance automotive transmissions commonly use these synchronizers.

Shift linkage translation along the negative direction causes the clutches to engage the ring with hub A. Shift linkage translation along the positive direction causes the clutches to engage the ring with hub B. When the magnitude of the shift linkage translation is smaller than the cone clutch ring-hub gap, the synchronizer is in neutral mode and does not transmit torque.

Cross-section of a double-sided synchronizer with the hub shafts on either side of the ring shaft. A translational detent holds the shift linkage in place.

The schematic illustrates a double-sided synchronizer in the disengaged state. In this state, the ring, R, and hub, H_A and H_B, shafts can spin independently at different speeds. As the shift linkage, S, translates in the negative direction, the faces of cone clutch A (CC_A) come into contact. The friction in the cone clutch decreases the difference in rotational speed between the shafts. When the force on the shift linkage exceeds the peak force of detent, D, the dog clutch teeth, T, can engage. The detent peak force should be such that the cone clutch has enough time and normal force to bring the shafts to sufficiently similar speeds to allow engagement of the dog clutch. Similarly, translating the shift linkage along the positive direction allows the faces of cone clutch B (CC_B) to come into contact, and can allow the shaft of the ring to engage with the shaft of the hub B (H_B).

Simscape model showing the equivalent composite block arrangement using two Dog Clutch blocks in parallel with two Cone Clutch blocks. A Translational Detent block is connected at the shift linkage between the Cone Clutch blocks and Dog Clutch blocks.

The model implements two Dog Clutch blocks, two Cone Clutch blocks, and one Translational Detent block. Refer to each block reference page for more information on the corresponding block function.

Connections R, H_A, and H_B are mechanical rotational conserving ports that represent the ring,R, hub A (H_A), and hub B (H_B), respectively. Connection S is a mechanical translational conserving port that represents the ring shifter handle.

Connections X1 and X2 are physical signal ports that output the shift linkage positions of the dog clutches and cone clutches, respectively. The tables provide the values of X1 and X2 in common clutch engagement cases.

Dog Clutch State	X1
Disengaged	0
Fully engaged with hub A	Negative sum of ring-hub gap and tooth height
Fully engaged with hub B	Positive sum of ring-hub gap and tooth height

Cone Clutch State	X2
Disengaged	0
Fully engaged with hub A	Negative value of ring-hub gap
Fully engaged with hub B	Positive value of ring-hub gap

The values of X1 and X2 are zero when the synchronizer is fully disengaged. When the dog clutch is fully engaged with hub A, X1 is equal to the negative sum of its ring-hub gap and tooth height. When the dog clutch is fully engaged with hub B, X1 is equal to the positive sum of its ring-hub gap and tooth height. When the cone clutch is fully engaged with hub A, X2 is equal to the negative of its ring-hub gap. When the cone clutch is fully engaged with hub B, X2 is equal to its ring-hub gap.

Dog Clutch Torque Transmission Models

You can choose from these torque transmission models.

Friction Clutch Approximate Model

When you set Torque transmission model to Friction clutch approximation - Suitable for HIL and, the block treats the clutch engagement as a friction phenomenon between the ring and the hub. This setting is better suited for linearization, fixed-step simulation, and hardware-in-loop (HIL) simulation. The block uses a composite implementation of the Fundamental Friction Clutch block.

When you use this setting, the clutch has three possible configurations: disengaged, engaged, and locked. When disengaged, the contact force between the ring and the hub is zero. This force remains zero until the shift linkage reaches the minimum position for engagement.

When the ring-hub tooth overlap, h, exceeds the minimum value for engagement, the contact force between the two components begins to increase linearly with the shift linkage position, z.

At full engagement, the contact force reaches its maximum value and the clutch state switches to locked. In this state, the ring and the hub spin as a unit without slip. To unlock the clutch, the transmitted torque must exceed the value of the Maximum transmitted torque parameter.

Dynamic with Backlash

When you set Torque transmission model to Dynamic with backlash, the block simulates clutch phenomena such as backlash, torsional compliance, and contact forces between ring and hub teeth. This model provides greater accuracy than the friction clutch approximation.

When you use this setting, the clutch has two possible configurations: disengaged and engaged. When disengaged, the contact force between the ring and the hub is zero. This force remains zero until the shift linkage reaches the minimum position for engagement.

When the ring-hub tooth overlap, h, exceeds the engagement threshold value, the clutch transmits torque. This torque is the sum of torsional spring and damper components, including backlash between the ring and hub teeth, such that $T_{C} = {\begin{matrix} - k_{R H} (ϕ - \frac{δ}{2}) - μ_{R} \cdot ω & ϕ > \frac{δ}{2} \\ 0 & - \frac{δ}{2} < ϕ < \frac{δ}{2} \\ - k_{R H} (ϕ + \frac{δ}{2}) - μ_{R} ω & ϕ < - \frac{δ}{2} \end{matrix},$ where:

k_RH is the torsional stiffness of the ring-hub coupling.
ϕ is the relative angle, about the common rotation axis, between the ring and the hub.
δ is the backlash between ring and hub teeth.
ω is the relative angular velocity between the ring and the hub. This variable describes how fast the two components slip past each other.

Compliant end stops limit the translational motion of the clutch shift linkage and the ring. The compliance model treats the end stops as linear spring-damper sets. The location of the end stops depends on the relative angle and angular velocity between the ring and hub teeth:

If the teeth align and the relative angular velocity is smaller than the maximum value for clutch engagement, the end stop location is the sum of the ring-hub clearance when fully disengaged and the tooth height. The clutch can engage in this end stop position.
If the teeth do not align or the relative angular velocity exceeds the maximum value for clutch engagement, the end-stop location is set to prevent the ring from engaging the hub. The clutch does not engage in this end stop position.

Translational friction opposes shift linkage and ring motion. This friction is the sum of Coulomb and viscous components, such that $F_{Z} = - k_{K} \cdot F_{N} \cdot \tanh (\frac{4 v}{v_{t h}}) - μ_{T} v,$ where:

F_Z is the net translational friction force acting on the shift linkage and ring.
k_K is the kinetic friction coefficient between ring and hub teeth.
F_N is the normal force between ring and hub teeth, where F_N = T_C/R_m.
v is the translational velocity of the shift linkage and the ring.
v_th is the translational velocity threshold. Below this threshold, a hyperbolic tangent function smooths the Coulomb friction force to zero as the shift linkage and ring velocity tends to zero.
μ_T is the viscous damping coefficient acting on the shift linkage and the ring.

Dynamic Modal

When you set Torque transmission model to Dynamic modal, the block determines the discrete clutch modal behavior by taking the shift linkage position from the mechanical translational conserving port S. This setting captures more complex clutch dynamics than the Two-mode parameterization and is faster than the Friction clutch approximation and Dynamic with backlash settings.

You can use the Dynamic modal setting to simulate engagement blocking when the speed difference is too large for the clutch plates to engage. The block represents engagement blocking as a spring-damper system where you can parameterize the spring and damping coefficients. The engagement modes overlap, which prevents the mode from changing until the shift linkage position is beyond the engagement overlap region. You define the engagement overlap region using the Tooth overlap to engage parameter.

Diagram depicting the mode transitions between engaged and disengage. The z variable represents the shift linkage position. The Engaged and Disengaged modal regions overlap, which attenuates the mode transition rate.

The linkage position constrains are:

z is the shift linkage position at port S.
h is the Tooth height parameter.
z_Gap is the Ring hub clearance when disengaged parameter.
z_Overlap is the Tooth overlap to engage parameter.
ω_thr is the Engagement speed threshold parameter.

The engagement mode positions are:

z = 0 — The clutch is fully disengaged.
0 < z < z_Gap — The clutch is disengaged when the shift linkage position is in the region defined by z_Gap. The clutch can transition from disengaged to engaged at x = z_Gap + z_Overlap.
z_Gap < z < h — The clutch is engaged when z is in the region defined by h. The clutch can transition from engaged to disengaged when z = z_Gap.
z = z_Gap + h — The clutch is fully engaged.
z = z_Gap + z_Overlap — When z is in the engagement overlap region, the clutch engages only when the speed difference is less than the value of the ω_thr parameter.

Thermal Modeling

You can model the effects of heat flow and temperature change through an optional thermal conserving port. By default, the thermal port is hidden. To expose the thermal port, in the Clutch settings, select a temperature-dependent setting tor the Friction model parameter. Specify the associated thermal parameters for the component.

Assumptions and Limitations

The model does not account for inertia effects. You can add a Simscape™ Inertia block at each port to add inertia to the synchronizer model.

Ports

Output

expand all

X1 — Dog clutch translation
physical signal

Physical signal output port that measures the magnitude of the dog clutch translation.

X2 — Cone clutch translation
physical signal

Physical signal output port that measures the magnitude of the cone clutch translation.

Conserving

expand all

HA — Clutch hub A
mechanical rotational

Mechanical rotational conserving port associated with the clutch hub A shaft.

HB — Clutch hub B
mechanical rotational

Mechanical rotational conserving port associated with the clutch hub B shaft.

R — Clutch ring
mechanical rotational

Mechanical rotational conserving port associated with the clutch ring.

S — Shift linkage
mechanical rotational

Mechanical rotational conserving port associated with shift linkage.

T — Heat flow
thermal

Thermal conserving port associated with heat flow.

Dependencies

To enable this parameter, in the Friction section, set Friction model to Temperature-dependent friction coefficients or Temperature and velocity-dependent friction coefficients.

Parameters

expand all

The table shows how the specified options for parameters in both the Cone Clutch and Dog Clutch settings affect the visibility of:

Parameters in the Cone Clutch, Dog Clutch, and Initial Conditions settings
Thermal Port settings
Thermal port T

Cone Clutch

Contact surface maximum diameter — Outer diameter
`150` `mm` (default) | positive scalar

Outer conical diameter d_o.

Contact surface minimum diameter — Inner diameter
`100` `mm` (default) | positive scalar

Inner conical diameter d_i.

Cone half angle — Cone half angle
`12` `deg` (default) | positive scalar

Half opening angle α of the cone geometry.

Friction model — Friction model
`Fixed kinetic friction coefficient` (default) | `Velocity-dependent kinetic friction coefficient` | `Temperature-dependent friction coefficients` | `Temperature and velocity-dependent friction coefficients`

Parameterization method to model the kinetic friction coefficient. The options and default values for this parameter depend on the friction model that you select for the block. The options are:

Fixed kinetic friction coefficient — Provide a fixed value for the kinetic friction coefficient.
Velocity-dependent kinetic friction coefficient — Define the kinetic friction coefficient by one-dimensional table lookup based on the relative angular velocity between disks.
Temperature-dependent friction coefficients — Define the kinetic friction coefficient by table lookup based on the temperature.
Temperature and velocity-dependent friction coefficients — Define the kinetic friction coefficient by table lookup based on the temperature and the relative angular velocity between disks.

Relative velocity vector — Relative velocity
`[0, 100, 1000]` `rad/s` (default) | vector

Input values for the relative velocity as a vector. The values in the vector must increase from left to right. The minimum number of values depends on the interpolation method that you select. For linear interpolation, provide at least two values per dimension. For smooth interpolation, provide at least three values per dimension.

Dependencies

To enable this parameter, set Friction model to Velocity-dependent kinetic friction coefficient or Temperature and velocity-dependent friction coefficients.

Temperature vector — Temperature
`[280, 300, 320]` `K` (default) | increasing vector

Input values for the temperature as a vector. The minimum number of values depends on the interpolation method that you select. For linear interpolation, provide at least two values per dimension. For smooth interpolation, provide at least three values per dimension. The values in the vector must increase from left to right.

Dependencies

To enable this parameter Friction model set Temperature-dependent friction coefficients or Temperature and velocity-dependent friction coefficients.

Static friction coefficient — Static friction coefficient
`0.35` (default) | scalar

Static or peak value of the friction coefficient. The static friction coefficient must be greater than the kinetic friction coefficient.

Dependencies

To enable this parameter, set Friction model to Fixed kinetic friction coefficient or Velocity-dependent kinetic friction coefficient.

Static friction coefficient vector — Static friction coefficient
`[.4, .38, .36]` (default) | vector

Static, or peak, values of the friction coefficient as a vector. The vector must have the same number of elements as the temperature vector. Each value must be greater than the value of the corresponding element in the kinetic friction coefficient vector.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent friction coefficients or Temperature and velocity-dependent friction coefficients.

Kinetic friction coefficient — Kinetic friction coefficient
`0.3` (default) | positive scalar

The kinetic, or Coulomb, friction coefficient. The coefficient must be greater than zero.

Dependencies

To enable this parameter, set Friction model to Fixed kinetic friction coefficient.

Kinetic friction coefficient vector — Kinetic friction coefficient
`[.3, .22, .19]` (default) | `[.3, .28, .25]` | vector

Output values for kinetic friction coefficient as a vector. All values must be greater than zero.

If the Friction model parameter is set to

Velocity-dependent kinetic friction coefficient — The vector must have same number of elements as relative velocity vector.
Temperature-dependent friction coefficients — The vector must have the same number of elements as the temperature vector.

Dependencies

To enable this parameter, set Friction model to Velocity-dependent kinetic friction coefficient or Temperature-dependent friction coefficients.

Kinetic friction coefficient matrix — Kinetic friction coefficient
`[.34, .32, .3; .3, .28, .25; .25, .2, .15]` (default) | matrix

Output values for kinetic friction coefficient as a matrix. All the values must be greater than zero. The size of the matrix must equal the size of the matrix that is the result of the temperature vector × the kinetic friction coefficient relative velocity vector.

Dependencies

To enable this parameter, set Friction model to Temperature and velocity-dependent friction coefficients.

Friction coefficient interpolation method — Interpolation method
`Linear` (default) | `Smooth`

Interpolation method for approximating the output value when the input value is between two consecutive grid points:

Linear — Select this option to get the best performance.
Smooth — Select this option to produce a continuous curve with continuous first-order derivatives.

For more information on interpolation algorithms, see the PS Lookup Table (1D) block reference page.

Dependencies

To enable this parameter, set Friction model to Velocity-dependent kinetic friction coefficient, Temperature-dependent friction coefficients, or Temperature and velocity-dependent friction coefficients.

Friction coefficient extrapolation method — Extrapolation method
`Linear` (default) | `Nearest` | `Error`

Extrapolation method for determining the output value when the input value is outside the range specified in the argument list:

Linear — Select this option to produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.
Nearest — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.
Error — Select this option to avoid going into the extrapolation mode when you want your data to be within the table range. If the input signal is outside the range of the table, the simulation stops and generates an error.

For more information on extrapolation algorithms, see the PS Lookup Table (1D) block reference page.

Dependencies

Velocity tolerance — Relative velocity locking threshold
`0.001` `rad/s` (default) | scalar

Relative velocity below which the two surfaces can lock. The surfaces lock if the torque is less than the product of the effective radius, the static friction coefficient, and the applied normal force.

Threshold force — Normal contact force threshold
`1` `N` (default) | scalar

Force threshold. The block simulates the normal force only when it exceeds the value of the Threshold force parameter. Otherwise, there is no transmitted frictional torque.

Dog Clutch

Torque transmission model — Torque transmission model
`Friction clutch approximation — Suitable for HIL and linearization` (default) | `Dynamic with backlash` | `Dynamic modal`

Method to use to model the behavior of the dog clutch:

Two-mode — Fully abstracted torque transmission model based on mode charts. This setting is fast enough for real time simulation and does not require knowledge of the clutch dimensions. You can control the shift linkage only with a physical or logic-controlled signal.
Friction clutch approximation - Suitable for HIL and linearization — Medium- to high-fidelity composite implementation of the Fundamental Friction Clutch block. This setting supports thermal modeling. You can control the shift linkage by using either a physical signal or a mechanical translational conserving connection.
Dynamic with backlash — High-fidelity clutch engagement model, that accounts for phenomena such as backlash, torsional compliance, and contact forces between the ring and hub teeth.
Dynamic modal — System-level clutch engagement model that captures the effects of engagement blocking due to the speed differential.

Temperature vector — Temperature
`[280, 300, 320]` `K` (default) | vector of increasing values

Temperature of the cone clutch. The minimum number of values depends on the interpolation method that you select. For linear interpolation, provide at least two values per dimension. For smooth interpolation, provide at least three values per dimension. The values in the vector must increase from left to right.

Dependencies

To enable this parameter, in the Cone Clutch section, set Friction model to Temperature-dependent friction coefficients or Temperature and velocity-dependent friction coefficients.

Maximum transmitted torque — Maximum transmitted torque
`1000` `N*m` (default) | positive scalar

Largest torque that the clutch can transmit, corresponding to a nonslip engaged configuration. If the torque transmitted between the ring and the hub exceeds this value, the two components begin to slip with respect to each other. This torque determines the static friction limit in the friction clutch approximation

Dependencies

To enable this parameter, in the Cone Clutch section, set Friction model to Fixed kinetic friction coefficient or Velocity-dependent kinetic friction coefficient and, in the Dog Clutch section, set Torque transmission model to Friction clutch approximation - Suitable for HIL and linearization.

Maximum transmitted torque vector — Maximum transmitted torque
`[1000, 1050, 1000]` `N*m` (default) | vector

Largest torque that the clutch can transmit, corresponding to a nonslip engaged configuration, specified as a vector. If the torque transmitted between the ring and the hub exceeds this value, the two components begin to slip with respect to each other. This torque determines the static friction limit in the friction clutch approximation. The vector has the same number of elements as the temperature vector.

Dependencies

To enable this parameter, in the Cone Clutch section, set Friction model to Temperature-dependent kinetic friction coefficient or Temperature and velocity-dependent kinetic friction coefficient.

Interpolation method — Interpolation method
`Linear` (default) | `Smooth`

Interpolation method for approximating the output value when the input value is between two consecutive grid points:

Linear — Select this option to get the best performance.
Smooth — Select this option to produce a continuous curve with continuous first-order derivatives.

For more information on interpolation algorithms, see the PS Lookup Table (1D) block reference page.

Dependencies

Extrapolation method — Extrapolation method
`Linear` (default) | `Nearest` | `Error`

Extrapolation method for determining the output value when the input value is outside the range specified in the argument list:

Linear — Select this option to produce a curve with continuous first-order derivatives in the extrapolation region and at the boundary with the interpolation region.
Nearest — Select this option to produce an extrapolation that does not go above the highest point in the data or below the lowest point in the data.
Error — Select this option to avoid going into the extrapolation mode when you want your data to be within the table range. If the input signal is outside the range of the table, the simulation stops and generates an error.

For more information on extrapolation algorithms, see the PS Lookup Table (1D) block reference page.

Dependencies

Clutch teeth mean radius — Clutch teeth mean radius
`50` `mm` (default) | positive scalar

Distance from the ring or hub center to the corresponding tooth center. The mean tooth radius determines the normal contact forces between ring and hub teeth given the transmission torque between the two components. The value of this parameter must be greater than zero.

Dependencies

Number of teeth — Number of ring or hub teeth
`6` (default) | positive nonzero scalar integer

Total number of teeth in the ring or the hub. The two components have equal tooth numbers. The value of this parameter must be greater than or equal to one.

Dependencies

Rotational backlash — Rotational backlash
`10` `deg` (default) | positive scalar

Allowable angular motion, or play, between the ring and hub teeth in the engaged clutch configuration. The value of this parameter must be greater than zero.

Dependencies

Torsional stiffness — Torsional stiffness
`10e6` `N*m/rad` (default) | positive scalar

Linear torsional stiffness coefficient at the contact interface between the ring and hub teeth. This coefficient characterizes the restoring component of the contact force between the two sets of teeth. Greater stiffness values correspond to greater contact forces. The value of this parameter must be greater than zero. The default value is 10e6 N*m/rad.

Dependencies

Torsional damping — Torsional damping
`100` `N*m/(rad/s)` (default) | positive scalar

Linear torsional damping coefficient at the contact interface between the ring and hub teeth. This coefficient characterizes the dissipative component of the contact force between the two sets of teeth. Greater damping values correspond to greater energy dissipation during contact. The value of this parameter must be greater than zero.

Dependencies

Detent

Peak force — Peak shear force
`500` `N` (default) | nonnegative scalar

Peak shear force of the detent.

Notch width — Notch width
`3` `mm` (default) | nonnegative scalar

Width of the region where the detent exhibits shear force.

Viscous friction coefficient — Viscous friction coefficient
`0.1` `N/(m/s)` (default) | nonnegative scalar

Viscous friction coefficient at the contact surface of the detent. The value of this parameter must be greater than or equal to zero.

Friction to peak force ratio — Friction to peak force ratio
`0.01` (default) | nonnegative scalar

Ratio of the kinetic friction to the peak shear force of the detent. The parameter is used to set the value of the kinetic friction. The parameter must be greater than or equal to zero.

Friction velocity threshold — Kinetic friction velocity threshold
`0.05` `m/s` (default)

Velocity required for peak kinetic friction at the contact surface of the detent. The parameter ensures the force is continuous when the travel direction changes, increasing the numerical stability of the simulation. The parameter must be greater than zero. The default value is 0.05 m/s.

Shift Linkage

Maximum engagement speed — Engagement speed upper velocity threshold
`inf` (default) | positive scalar

Relative angular velocity between the ring and the hub above which the clutch cannot engage. The value is specific to the specific gearbox or transmission. Minimizing the value helps avoid high dynamic impact during engagement. The value of this parameter must be greater than zero.

Tooth overlap to engage — Tooth overlap engagement threshold
`3` `mm` (default) | positive scalar

Overlap length between the ring and hub teeth along the common longitudinal axis. The clutch engages when the tooth overlap is greater than this value. The clutch remains disengaged until the meshing gear teeth overlap by at least this length. The value must be greater than zero.

Tooth height — Tooth height
`10` `mm` (default) | positive scalar

Distance between the base and crest of a tooth. Ring and hub teeth share the same height. The tooth height and the ring-hub clearance when fully disengaged determine the maximum travel span of the shift linkage. The value of this parameter must be greater than zero.

Ring-hub clearance when disengaged — Ring-hub clearance when disengaged
`3` `mm` (default) | positive scalar

Maximum open gap between the ring and hub tooth crests along the shift linkage translation axis. This gap corresponds to the fully disengaged clutch state. The tooth height and the ring-hub clearance when fully disengaged determine the maximum travel span of the shift linkage. The value of this parameter must be greater than zero.

Hard stop at back of shift linkage — Hard stop model
`On` (default) | `Off`

Hard stop that prevents the shift linkage from traveling beyond the fully disengaged position:

On — Hard stop when fully disengaged.
Off — No hard stop when fully disengaged.

Dog clutch ring stop stiffness — Dog clutch ring stop stiffness
`10e5` `N/m` (default) | positive scalar

Stiffness of the hard stops on both sides of the dog clutch ring. The model assumes the ring and stops behave elastically. Contact deformation is proportional to the applied force and the reciprocal of the contact stiffness. The value of the stiffness must be assigned with reference to the parameter Tooth overlap to engage. Too low a stiffness could cause the deformation to exceed the required overlap and initiate a false engagement. The parameter must be greater than zero.

Cone clutch ring stop stiffness — Cone clutch ring stop stiffness
`10e5` `N/m` (default) | positive scalar

Stiffness of the hard stops on both sides of the cone clutch ring. The model assumes the ring and stops behave elastically. Contact deformation is proportional to the applied force and the reciprocal of the contact stiffness.

Dog clutch ring stop damping — Dog clutch ring stop damping
`1e3` `N/(m/s)` (default) | nonegative scalar

Translational contact damping between the dog clutch ring and the hub. The value of the damping is inversely proportional to the number of oscillations that occur after impact. The parameter must be greater than zero.

Cone clutch ring stop damping — Cone clutch ring stop damping
`1e3` `N/(m/s)` (default) | nonegative scalar

Translational contact damping between the cone clutch ring and the hub. The value of damping is inversely proportional to the number of oscillations that occur after impact. The parameter must be greater than zero.

Shift linkage viscous friction coefficient — Shift linkage viscous friction coefficient
`100` `N/(m/s)` (default) | positive scalar

Viscous friction coefficient for the relative translational motion between the hub and the ring. The value of the parameter depends on lubrication state and quality of contacting surfaces. The coefficient must be greater than or equal to zero.